Two-dimensional gauge theory and matrix model

We study a matrix model obtained by dimensionally reducing Chern–Simons theory on S³$S^3$. We find that the matrix integration is decomposed into sectors classified by the representation of SU(2)$\mathit{SU}(2)$. We show that the N  -block sectors reproduce SU(N)$\mathit{SU}(N)$ Yang–Mills theory on S²$S^2$ as the matrix size goes to infinity.     

Composite representation invariants and unoriented topological string amplitudes

Sinha and Vafa [1] had conjectured that the SO   Chern–Simons gauge theory on S³$S^3$ must be dual to the closed A  -model topological string on the orientifold of a resolved conifold. Though the Chern–Simons free energy could be rewritten in terms of the topological string amplitudes providing evidence for the conjecture, we needed a novel idea in the context of Wilson loop observables to extract cross-cap c=0,1,2$c=0,1,2$ topological amplitudes. Recent paper of Marino [2] based on the work of Morton and Ryder [3] has clearly shown that the composite representation placed on the knots and links plays a crucial role to rewrite the topological string cross-cap c=0$c=0$ amplitude. This enables extracting the unoriented cross-cap c=2$c=2$ topological amplitude. In this paper, we have explicitly worked out the composite invariants for some framed knots and links carrying composite representations in U(N)$U(N)$ Chern–Simons theory. We have verified generalised Rudolph's theorem, which relates composite invariants to the invariants in SO(N)$\mathit{SO}(N)$ Chern–Simons theory, and also verified Marino's conjectures on the integrality properties of the topological string amplitudes. For some framed knots and links, we have tabulated the BPS integer invariants for cross-cap c=0$c=0$ and c=2$c=2$ giving the open-string topological amplitude on the orientifold of the resolved conifold.     

On DBI Textures with generalized Hopf fibration

In this Letter we show numerical existence of O(4)$O(4)$ Dirac–Born–Infeld (DBI) Textures living in (N+1)$(N+1)$ dimensional spacetime. These defects are characterized by S^N→S³$S^N\to S^3$ mapping, generalizing the well-known Hopf fibration into _πN(S³)${\pi }_N(S^3)$, for all N>3$N>3$. The nonlinear nature of DBI kinetic term provides stability against size perturbation and thus renders the defects having natural scale.     

Eikonal contributions to ultra high energy neutrino–nucleon cross sections in low scale gravity models

We calculate low scale gravity effects on the cross section for neutrino–nucleon scattering at center of mass energies up to the Greisen–Zatsepin–Kuzmin (GZK) scale, in the eikonal approximation. We compare the cases of an infinitely thin brane embedded in n=5$n=5$ compactified extra-dimensions, and of a brane with a physical tension MS=1 TeV$M_S=1\text{TeV}$ and MS=10 TeV$M_S=10\text{TeV}$. The extra dimensional Planck scale MD$M_D$ is set at ¹⁰³ GeV${10}^3\text{GeV}$ and 2×¹⁰³ GeV$2\times {10}^3\text{GeV}$. We also compare our calculations with neutral current standard model calculations in the same energy range, and compare the thin brane eikonal cross section to its saddle point approximation. New physics effects enhance the cross section by orders of magnitude on average. They are quite sensitive to MS$M_S$ and MD$M_D$ choices, though much less sensitive to n.     

Bertrand curves in the three-dimensional sphere

A curve α$\alpha$ immersed in the three-dimensional sphere S³${\mathbb{S}}^3$ is said to be a Bertrand curve if there exists another curve β$\beta$ and a one-to-one correspondence between α$\alpha$ and β$\beta$ such that both curves have common principal normal geodesics at corresponding points. The curves α$\alpha$ and β$\beta$ are said to be a pair of Bertrand curves in S³${\mathbb{S}}^3$. One of our main results is a sort of theorem for Bertrand curves in S³${\mathbb{S}}^3$ which formally agrees with the classical one: “Bertrand curves in S³${\mathbb{S}}^3$ correspond to curves for which there exist two constants λ≠0$\lambda \ne 0$ and μ$\mu$ such that λκ+μτ=1$\lambda \kappa +\mu \tau =1$”, where κ$\kappa$ and τ$\tau$ stand for the curvature and torsion of the curve; in particular, general helices in the 3-sphere introduced by M. Barros are Bertrand curves. As an easy application of the main theorem, we characterize helices in S³${\mathbb{S}}^3$ as the only twisted curves in S³${\mathbb{S}}^3$ having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in S³${\mathbb{S}}^3$ and (1,3)-Bertrand curves in R⁴${\mathbb{R}}^4$.     

The puzzle of apparent linear lattice artifacts in the 2d non-linear σ-model and Symanzik's solution

Lattice artifacts in the 2d O(n) non-linear σ  -model are expected to be of the form O(a²)$\mathrm{O}(a^2)$, and hence it was (when first observed) disturbing that some quantities in the O(3)$\mathrm{O}(3)$ model with various actions show parametrically stronger cutoff dependence, apparently O(a)$\mathrm{O}(a)$, up to very large correlation lengths. In a previous letter Balog et al. (2009) [1] we described the solution to this puzzle. Based on the conventional framework of Symanzik's effective action, we showed that there are logarithmic corrections to the O(a²)$\mathrm{O}(a^2)$ artifacts which are especially large (ln³a${\ln }^{3}a$) for n=3$n=3$ and that such artifacts are consistent with the data. In this paper we supply the technical details of this computation. Results of Monte Carlo simulations using various lattice actions for O(3)$\mathrm{O}(3)$ and O(4)$\mathrm{O}(4)$ are also presented.     

Chaotic inflation,radiative corrections and precision cosmology

We employ chaotic (?²${?}^2$ and ?⁴${?}^4$) inflation to illustrate the important role radiative corrections can play during the inflationary phase. Yukawa interactions of ?  , in particular, lead to corrections of the form −κ?⁴ln(?/μ)$-\kappa {?}^4\ln (?/\mu )$, where κ>0$\kappa >0$ and μ   is a renormalization scale. For instance, ?⁴${?}^4$ chaotic inflation with radiative corrections looks compatible with the most recent WMAP (5 year) analysis, in sharp contrast to the tree level case. We obtain the 95% confidence limits 2.4×¹⁰⁻¹⁴?κ?5.7×¹⁰⁻¹⁴$2.4\times {10}^{\mathrm{-14}}?\kappa ?5.7\times {10}^{\mathrm{-14}}$, 0.931?ns?0.958$0.931?n_s?0.958$ and 0.038?r?0.205$0.038?r?0.205$, where ns$n_s$ and r   respectively denote the scalar spectral index and scalar to tensor ratio. The limits for ?²${?}^2$ inflation are κ?7.7×¹⁰⁻¹⁵$\kappa ?7.7\times {10}^{\mathrm{-15}}$, 0.929?ns?0.966$0.929?n_s?0.966$ and 0.023?r?0.135$0.023?r?0.135$. The next round of precision experiments should provide a more stringent test of realistic chaotic ?²${?}^2$ and ?⁴${?}^4$ inflation.     

Galilean anti-de-Sitter spacetime in Romans theory

The Romans type IIA theory is the only known example of 10-dimensional maximal supergravity where (tensor) fields are explicitly massive. We provide an example of a non-relativistic anti-de-Sitter NRadS₄×S⁶${\mathit{NRadS}}_4\times S^6$ background as a solution in massive type IIA. A compactification of which on S⁶$S^6$ gives immediately the prototype NRadS background in D=4$D=4$ which is proposed to be dual to ‘cold atoms’ or unitary fermions on a wire.     

On analogues of black brane solutions in the model with multicomponent anisotropic fluid

A family of spherically symmetric solutions with horizon in the model with m  -component anisotropic fluid is presented. The metrics are defined on a manifold that contains a product of n−1$n-1$ Ricci-flat “internal” spaces. The equation of state for any s  -th component is defined by a vector Us$U^s$ belonging to Rn+1${\mathbb{R}}^{n+1}$. The solutions are governed by moduli functions Hs$H_s$ obeying non-linear differential equations with certain boundary conditions imposed. A simulation of black brane solutions in the model with antisymmetric forms is considered. An example of solution imitating M₂–M₅$M_2\text{–}{M}_5$ configuration (in D=11$D=11$ supergravity) corresponding to Lie algebra A₂$A_2$ is presented.     

Holographic cosmological models on the braneworld

In this Letter we have studied a closed universe which a holographic energy on the brane whose energy density is described by ρ(H)=3c²H²$\rho (H)=3c^2{H}^2$ and we obtain an equation for the Hubble parameter. This equation gave us different physical behavior depending if c²>1$c^2>1$ or c²<1$c^2<1$ against of the sign of the brane tension.     

The nature of ZZ branes

In minimal non-critical string theory we show that the principal (r,s)$(r,s)$ ZZ brane can be viewed as the basic (1,1)$(1,1)$ ZZ boundary state tensored with the (r,s)$(r,s)$ Cardy boundary state. In this sense there exists only one ZZ boundary state, the basic (1,1)$(1,1)$ boundary state.     

A Monte Carlo model for networks between professionals and society

We propose a network model with a fixed number of nodes and links and with a dynamic which favors links between nodes differing in connectivity. We observe a phase transition and parameter regimes with degree distributions following power laws, P(k)∼k^-γ$P(k)\sim k^{-\gamma }$, with γ$\gamma$ ranging from 0.2$0.2$ to 0.5$0.5$, small-world properties, with a network diameter following D(N)∼logN$D(N)\sim \log N$ and relative high clustering, following C(N)∼1/N$C(N)\sim 1/N$ and C(k)∼k^-α$C(k)\sim k^{-\alpha }$, with α$\alpha$ close to 3. We compare our results with data from real-world protein interaction networks.     

Understanding the radiative decays of vector charmonia to light pseudoscalar mesons

We show that the newly measured branching ratios of vector charmonia (J/ψ$J/\psi$, ψ^′${\psi }^{\prime }$ and ψ(3770)$\psi (3770)$) into γP, where P   stands for light pseudoscalar mesons π⁰${\pi }^0$, η  , and η^′${\eta }^{\prime }$, can be well understood in the framework of vector meson dominance (VMD) in association with the ηc–η(η^′)${\eta }_c\text{–}\eta ({\eta }^{\prime })$ mixings due to the axial gluonic anomaly. These two mechanisms behave differently in J/ψ$J/\psi$ and ψ^′→γP${\psi }^{\prime }\to \gamma P$. A coherent understanding of the branching ratio patterns observed in J/ψ(ψ^′)→γP$J/\psi ({\psi }^{\prime })\to \gamma P$ can be achieved by self-consistently including those transition mechanisms at hadronic level. The branching ratios for ψ(3770)→γP$\psi (3770)\to \gamma P$ are predicted to be rather small.     

Neumann-like integrable models

A countable class of integrable dynamical systems, with four-dimensional phase space and conserved quantities in involution (Hn,In)$(H_n,I_n)$ are exhibited. For n=1$n=1$ we recover Neumann system on T^∗S²$T^{\ast }{S}^2$. All these systems are also integrable at the quantum level.     

Minimal hypersurfaces in a nearly Kaehler 6-sphere

In this paper we show that for a compact minimal hypersurface M$M$ of constant scalar curvature in the unit sphere S⁶$S^6$ with the shape operator A$A$ satisfying ‖A‖²>5${\Vert A\Vert }^2>5$, there exists an eigenvalue λ>10$\lambda >10$ of the Laplace operator of the hypersurface M$M$ such that ‖A‖²=λ−5${\Vert A\Vert }^2=\lambda -5$. This gives the next discrete value of ‖A‖²${\Vert A\Vert }^2$ greater than 0 and 5.